RVineGrad {VineCopula}

Gradient of the Log-Likelihood of an R-Vine Copula Model

Description

This function calculates the gradient of the log-likelihood of a d-dimensional R-vine copula model with respect to the copula parameter and evaluates it on a given copula data set.

Usage

RVineGrad(
  data,
  RVM,
  par = RVM$par,
  par2 = RVM$par2,
  start.V = NA,
  posParams = (RVM$family > 0)
)

Arguments

data

An N x d data matrix (with uniform margins).

RVM

An RVineMatrix() object including the structure and the pair-copula families and parameters.
Only the following copula families are allowed in RVM$family
0 = independence copula
1 = Gaussian copula
2 = Student t copula (t-copula)
3 = Clayton copula
4 = Gumbel copula
5 = Frank copula
6 = Joe copula
13 = rotated Clayton copula (180 degrees; ⁠survival Clayton'') \cr `14` = rotated Gumbel copula (180 degrees; ⁠survival Gumbel”)
16 = rotated Joe copula (180 degrees; “survival Joe”)
23 = rotated Clayton copula (90 degrees)
'24' = rotated Gumbel copula (90 degrees)
'26' = rotated Joe copula (90 degrees)
'33' = rotated Clayton copula (270 degrees)
'34' = rotated Gumbel copula (270 degrees)
'36' = rotated Joe copula (270 degrees)

par

A d x d matrix with the pair-copula parameters (optional; default: par = RVM$par).

par2

A d x d matrix with the second parameters of pair-copula families with two parameters (optional; default: par2 = RVM$par2).

start.V

Transformations (h-functions and log-likelihoods of each pair-copula) of previous calculations (see output; default: start.V = NA).

posParams

A d x d matrix indicating which copula has to be considered in the gradient (default: posParams = (RVM$family > 0)).

Details

The ordering of the gradient is due to the ordering of the R-vine matrix. The gradient starts at the lower right corner of the R-vine matrix and goes column by column to the left and up, i.e. the first entry of the gradient is the last entry of the second last column of the par-matrix followed by the last entry of the third last column and the second last entry of this column. If there is a copula family with two parameters, i.e. the t-copula, the derivative with respect to the second parameter is at the end of the gradient vector in order of their occurrence.

Value

gradient The calculated gradient of the log-likelihood value of the R-vine copula model. (three matrices: direct, indirect and value).

Note

The gradient for R-vine copula models with two parameter Archimedean copulas, i.e. BB1, BB6, BB7, BB8 and their rotated versions can not yet be calculated. The derivatives of these bivariate copulas are more complicated.

Author(s)

Ulf Schepsmeier, Jakob Stoeber

References

Dissmann, J. F., E. C. Brechmann, C. Czado, and D. Kurowicka (2013). Selecting and estimating regular vine copulae and application to financial returns. Computational Statistics & Data Analysis, 59 (1), 52-69.

Schepsmeier, U. and J. Stoeber (2014) Derivatives and Fisher information of bivariate copulas. Statistical Papers, 55(2), 525-542. online first: https://link.springer.com/article/10.1007/s00362-013-0498-x.

Web supplement: Derivatives and Fisher Information of bivariate copulas. https://mediatum.ub.tum.de/node?id=1119201

Stoeber, J. and U. Schepsmeier (2013). Estimating standard errors in regular vine copula models. Computational Statistics, 28 (6), 2679-2707 https://link.springer.com/article/10.1007/s00180-013-0423-8#.

See Also

BiCopDeriv(), BiCopDeriv2(), BiCopHfuncDeriv(), BiCopHfuncDeriv2(),
RVineMatrix(), RVineMLE(), RVineHessian()

Examples

# define 5-dimensional R-vine tree structure matrix
Matrix <- c(5, 2, 3, 1, 4,
            0, 2, 3, 4, 1,
            0, 0, 3, 4, 1,
            0, 0, 0, 4, 1,
            0, 0, 0, 0, 1)
Matrix <- matrix(Matrix, 5, 5)

# define R-vine pair-copula family matrix
family <- c(0, 1, 3, 4, 4,
            0, 0, 3, 4, 1,
            0, 0, 0, 4, 1,
            0, 0, 0, 0, 3,
            0, 0, 0, 0, 0)
family <- matrix(family, 5, 5)

# define R-vine pair-copula parameter matrix
par <- c(0, 0.2, 0.9, 1.5, 3.9,
         0, 0, 1.1, 1.6, 0.9,
         0, 0, 0, 1.9, 0.5,
         0, 0, 0, 0, 4.8,
         0, 0, 0, 0, 0)
par <- matrix(par, 5, 5)

# define second R-vine pair-copula parameter matrix
par2 <- matrix(0, 5, 5)

# define RVineMatrix object
RVM <- RVineMatrix(Matrix = Matrix, family = family,
                   par = par, par2 = par2,
                   names = c("V1", "V2", "V3", "V4", "V5"))

# simulate a sample of size 300 from the R-vine copula model
set.seed(123)
simdata <- RVineSim(300, RVM)

# compute the gradient of the first row of the data
out2 <- RVineGrad(simdata[1,], RVM)
out2$gradient